A $180^\circ$ rotation around the origin in the counter-clockwise direction is applied to $-6 - 3i.$  What is the resulting complex number?
Explanation: A $180^\circ$ rotation in the counter-clockwise direction corresponds to multiplication by $\operatorname{cis} 180^\circ = -1.$

[asy]
unitsize(0.5 cm);

pair A = (-6,-3), B = (6,3);

draw((-8,0)--(8,0));
draw((0,-4)--(0,4));
draw((0,0)--A,dashed);
draw((0,0)--B,dashed);

dot("$-6 - 3i$", A, SW);
dot("$6 + 3i$", B,  NE);
[/asy]

Hence, the image of $-6 - 3i$ is $(-1)(-6 - 3i) = \boxed{6 + 3i}.$